Stability and Error Analysis for the First-Order Euler Implicit/Explicit Scheme for the 3D MHD Equations

Abstract

This paper focuses on the stability and convergence analysis of the first-order Euler implicit/explicit scheme based on mixed finite element approximation for three-dimensional (3D) time-dependent MHD equations. Firstly, for initial data [Formula: see text] with [Formula: see text], the regularity results of the continuous solution [Formula: see text] and the spatial semi-discretization solution [Formula: see text] are obtained, and [Formula: see text]-error estimate of [Formula: see text] is deduced by using the negative norm technique. Next, through the use of mathematic induction, the [Formula: see text]-stability of the fully discrete first-order scheme is proved under the stability condition depending on the smoothness of initial data. Here, the stability condition is [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] where [Formula: see text] is some positive constant. Then, under the stability condition, the optimal [Formula: see text]-[Formula: see text] error estimate of the fully discrete solution [Formula: see text] and optimal [Formula: see text]-error estimate of the fully discrete solution [Formula: see text] are established by using the parabolic dual argument.